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Using egg cartons to hold the eggs seems like a good idea, but I'm not quit sure what you mean by "alternating which side of the carton is leaned up on something" ?   How are you using the cartons...just one side, and remove the top ?
What are you leaning the carton against ?   Using egg cartons the eggs would face vertical, I've been reading the eggs should lie horizontal....so maybe you can elaborate a little on you egg carton hatching technique...and of course a picture usually can answer a lot of questions.
I've looked, and it seems that i don't have a picture of my set-up.  I would take one, but my chicks are in mid-hatch right now, so that will have to wait.  Let me see if i can describe my set-up clearly.
I have found that it works well in this size incubator to incubate 24 eggs at once.  I cut the tops off of two of the paper/cardboard egg cartons to use the bottoms to hold eggs.  You can either leave them whole or cut them apart so you have 4 sections of 6 spots each.  I have done it both ways.  The eggs are placed vertically, fat side up, in the egg cartons.  
In the middle of the incubator, i place one of the carton tops.  This is the "something" i use to lean my egg cartons on.  So basically, if i have two egg carton bottoms, holding 12 eggs each, i will prop one long side of each egg carton bottom up on the egg carton top that is in the middle of the 'bator.  Place them so that the eggs are turned at roughly a 45 degree angle.  When it's time to "turn" the eggs, just switch the positions of the two egg carton bottoms.  Then they will be leaning in an opposite direction.
I put the wet rags that i use for hydration close to the edges of the inside of the 'bator, on top of the grate, so that they're easy to access but don't get the egg cartons wet.  I set my hygrometer on the egg carton top that is in the middle, and i balance my tiny thermometer on top of the eggs.
When it comes time for lock down, i remove the egg carton top from the bottom, place the egg carton bottoms, full of eggs, against the walls of the 'bator and move the hydration cloths to the middle.  This way, if i feel it's getting to dry in there, i can pour water onto the rags, through one of the vent holes in the lid, and i don't have to open the incubator.
As to the position of the eggs, it is my understanding that the big hatcheries use egg turners that keep the eggs in this same position, as do many automatic egg turners that you can get for our little home incubators.  It does seem helpful (just from my reading here on byc) that you keep the eggs in the same position during incubation as you wish to hatch them in.  Meaning, if you want to hatch in cartons, incubate in cartons - and vice versa.  Switching positions for hatch (from vertical to horizontal or from horizontal to vertical) appears to sometimes cause problems.
One other thing i do, that you didn't really ask about, is to put some of the puffy shelf covering (the kind that's kind of full of holes) on top of the grate.  This helps with clean-up, since it's machine washable, and it's also nicer for the chicks to walk on than the grate.
I hope this helps.  
I'm happy to answer any questions.  I hope i described this well.  A picture would definitely help.  I'll try to remedy that after this hatch is over.